A processing machine with redundant actuators includes multiple actuators for jointly positioning a worktool along each axis of motion. The position of the worktool is the algebraic sum of the position of the slow actuator and fast actuator positions. Thus, the machine is over-actuated, and degrees of freedom are available to optimize the movement of the worktool along a desired processing pattern. The worktool can be positioned by independent operations of the redundant actuators, and thus the task of positioning the worktool along the processing pattern can be separated between redundant actuators. One example of such a machine is a laser processing machine with redundant actuators.
Some conventional methods, see, e.g., U.S. Pat. Nos. 5,452,275 and 7,710,060, use frequency separation techniques to assign the task of positioning the laser beam to two actuators. For example, the processing pattern is filtered by a low pass filter. The filtered signal becomes a reference trajectory for one actuator, while a difference between the processing pattern and the filtered signal becomes a reference trajectory for another actuator. However, the filtering does not consider various constraints of the actuators, such as constraints on the accelerations or velocities. Furthermore, there is no guarantee that the separation in frequencies provides the optimal reference trajectories.
One method described in U.S. Publication 2013/0190898 generates the reference trajectories that account for the constraints based on Model Predictive Control (MPC). MPC is based on an iterative, finite horizon optimization of a model of a machine and has the ability to anticipate future events to take appropriate control actions. This is achieved by optimizing the operation of the machine over a future finite time-horizon subject to constraints, and only implementing the control over the current timeslot. For example, the constraints can represent physical limitation of the machine, legitimate and safety limitations on the operation of the machine, and performance limitations on a trajectory. A control strategy for the machine is admissible when the motion generated by the machine for such a control strategy satisfies all the constraints.
For example, at a current time t, a state of the machine is sampled and an admissible cost minimizing control strategy is determined for a relatively short time horizon in the future. Specifically, an online or on-the-fly calculation determines a cost-minimizing control strategy until a future time t+T. Only the first step of the control strategy is implemented, then the state is sampled again and the calculations are repeated starting from the new current state, yielding a new control and new predicted state path. The prediction horizon is continuously shifted forward. For this reason, MPC is also called receding horizon control.
However, due to the tracking nature of the MPC in this problem, such a receding horizon control approach has no guarantee, in general, of finding a solution to the optimization problem. Due to the receding horizon nature of the finite horizon optimal control problem, the existence of the solution for a certain window of data (horizon) does not by itself guarantee that when the window is shifted forward in time, a solution still exists.
One method solves this MPC problem using an initial feasible reference trajectory and an additional terminal equality constraint. That method imposes coincidence of the MPC-generated trajectory with the reference trajectory at specific instants of time, namely the end of each data window, and uses the remaining degrees of freedom to optimize the trajectory performance. However, the loss of performance caused by the terminal equality constraint is significant, and a long horizon required for this method results in significant computational complexity.
Accordingly, there is a need for a method for controlling an operation of a processing machine with redundant actuators that guarantees a priori satisfaction of constraints of the operation for processing variety of patterns.